Theorem ce2le 6234

Description: Partial ordering law for base two cardinal exponentiation. Theorem 4.8 of [Specker] p. 973. (Contributed by SF, 16-Mar-2015.)

Assertion

Ref	Expression
ce2le	⊢ (((M ∈ NC ∧ N ∈ NC ∧ (N ↑c 0c) ∈ NC ) ∧ M ≤c N) → (2c ↑c M) ≤c (2c ↑c N))

Proof of Theorem ce2le

Dummy variables y p q r s x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ce0t 6233	. . . 4 ⊢ ((N ∈ NC ∧ (N ↑c 0c) ∈ NC ) → ∃p ∈ NC N = Tc p)
2	1	3adant1 973	. . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ∧ (N ↑c 0c) ∈ NC ) → ∃p ∈ NC N = Tc p)
3	2	adantr 451	. 2 ⊢ (((M ∈ NC ∧ N ∈ NC ∧ (N ↑c 0c) ∈ NC ) ∧ M ≤c N) → ∃p ∈ NC N = Tc p)
4		letc 6232	. . . . . . . . 9 ⊢ ((M ∈ NC ∧ p ∈ NC ∧ M ≤c Tc p) → ∃q ∈ NC M = Tc q)
5		tlecg 6231	. . . . . . . . . . . . . . . . 17 ⊢ ((q ∈ NC ∧ p ∈ NC ) → (q ≤c p ↔ Tc q ≤c Tc p))
6	5	ancoms 439	. . . . . . . . . . . . . . . 16 ⊢ ((p ∈ NC ∧ q ∈ NC ) → (q ≤c p ↔ Tc q ≤c Tc p))
7		elncs 6120	. . . . . . . . . . . . . . . . . . . 20 ⊢ (q ∈ NC ↔ ∃x q = Nc x)
8		elncs 6120	. . . . . . . . . . . . . . . . . . . 20 ⊢ (p ∈ NC ↔ ∃y p = Nc y)
9	7, 8	anbi12i 678	. . . . . . . . . . . . . . . . . . 19 ⊢ ((q ∈ NC ∧ p ∈ NC ) ↔ (∃x q = Nc x ∧ ∃y p = Nc y))
10		eeanv 1913	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃x∃y(q = Nc x ∧ p = Nc y) ↔ (∃x q = Nc x ∧ ∃y p = Nc y))
11	9, 10	bitr4i 243	. . . . . . . . . . . . . . . . . 18 ⊢ ((q ∈ NC ∧ p ∈ NC ) ↔ ∃x∃y(q = Nc x ∧ p = Nc y))
12		enpw 6088	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (p ≈ x → ℘p ≈ ℘x)
13		elnc 6126	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (p ∈ Nc x ↔ p ≈ x)
14		elnc 6126	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (℘p ∈ Nc ℘x ↔ ℘p ≈ ℘x)
15	12, 13, 14	3imtr4i 257	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (p ∈ Nc x → ℘p ∈ Nc ℘x)
16	15	adantr 451	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((p ∈ Nc x ∧ q ∈ Nc y) → ℘p ∈ Nc ℘x)
17	16	adantr 451	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((p ∈ Nc x ∧ q ∈ Nc y) ∧ p ⊆ q) → ℘p ∈ Nc ℘x)
18		enpw 6088	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (q ≈ y → ℘q ≈ ℘y)
19		elnc 6126	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (q ∈ Nc y ↔ q ≈ y)
20		elnc 6126	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (℘q ∈ Nc ℘y ↔ ℘q ≈ ℘y)
21	18, 19, 20	3imtr4i 257	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (q ∈ Nc y → ℘q ∈ Nc ℘y)
22	21	adantl 452	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((p ∈ Nc x ∧ q ∈ Nc y) → ℘q ∈ Nc ℘y)
23	22	adantr 451	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((p ∈ Nc x ∧ q ∈ Nc y) ∧ p ⊆ q) → ℘q ∈ Nc ℘y)
24		sspwb 4119	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (p ⊆ q ↔ ℘p ⊆ ℘q)
25	24	biimpi 186	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (p ⊆ q → ℘p ⊆ ℘q)
26	25	adantl 452	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((p ∈ Nc x ∧ q ∈ Nc y) ∧ p ⊆ q) → ℘p ⊆ ℘q)
27		sseq1 3293	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (r = ℘p → (r ⊆ s ↔ ℘p ⊆ s))
28		sseq2 3294	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (s = ℘q → (℘p ⊆ s ↔ ℘p ⊆ ℘q))
29	27, 28	rspc2ev 2964	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((℘p ∈ Nc ℘x ∧ ℘q ∈ Nc ℘y ∧ ℘p ⊆ ℘q) → ∃r ∈ Nc ℘x∃s ∈ Nc ℘yr ⊆ s)
30	17, 23, 26, 29	syl3anc 1182	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((p ∈ Nc x ∧ q ∈ Nc y) ∧ p ⊆ q) → ∃r ∈ Nc ℘x∃s ∈ Nc ℘yr ⊆ s)
31	30	ex 423	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((p ∈ Nc x ∧ q ∈ Nc y) → (p ⊆ q → ∃r ∈ Nc ℘x∃s ∈ Nc ℘yr ⊆ s))
32	31	rexlimivv 2744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃p ∈ Nc x∃q ∈ Nc yp ⊆ q → ∃r ∈ Nc ℘x∃s ∈ Nc ℘yr ⊆ s)
33		ncex 6118	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Nc x ∈ V
34		ncex 6118	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Nc y ∈ V
35	33, 34	brlec 6114	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( Nc x ≤c Nc y ↔ ∃p ∈ Nc x∃q ∈ Nc yp ⊆ q)
36		ncex 6118	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Nc ℘x ∈ V
37		ncex 6118	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Nc ℘y ∈ V
38	36, 37	brlec 6114	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( Nc ℘x ≤c Nc ℘y ↔ ∃r ∈ Nc ℘x∃s ∈ Nc ℘yr ⊆ s)
39	32, 35, 38	3imtr4i 257	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( Nc x ≤c Nc y → Nc ℘x ≤c Nc ℘y)
40		vex 2863	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ x ∈ V
41	40	tcnc 6226	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Tc Nc x = Nc ℘1x
42	40	ce2 6193	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( Tc Nc x = Nc ℘1x → (2c ↑c Tc Nc x) = Nc ℘x)
43	41, 42	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2c ↑c Tc Nc x) = Nc ℘x
44		vex 2863	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ y ∈ V
45	44	tcnc 6226	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Tc Nc y = Nc ℘1y
46	44	ce2 6193	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( Tc Nc y = Nc ℘1y → (2c ↑c Tc Nc y) = Nc ℘y)
47	45, 46	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2c ↑c Tc Nc y) = Nc ℘y
48	39, 43, 47	3brtr4g 4672	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( Nc x ≤c Nc y → (2c ↑c Tc Nc x) ≤c (2c ↑c Tc Nc y))
49		breq12 4645	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((q = Nc x ∧ p = Nc y) → (q ≤c p ↔ Nc x ≤c Nc y))
50		tceq 6159	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (q = Nc x → Tc q = Tc Nc x)
51	50	oveq2d 5539	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (q = Nc x → (2c ↑c Tc q) = (2c ↑c Tc Nc x))
52		tceq 6159	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (p = Nc y → Tc p = Tc Nc y)
53	52	oveq2d 5539	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (p = Nc y → (2c ↑c Tc p) = (2c ↑c Tc Nc y))
54	51, 53	breqan12d 4655	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((q = Nc x ∧ p = Nc y) → ((2c ↑c Tc q) ≤c (2c ↑c Tc p) ↔ (2c ↑c Tc Nc x) ≤c (2c ↑c Tc Nc y)))
55	49, 54	imbi12d 311	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((q = Nc x ∧ p = Nc y) → ((q ≤c p → (2c ↑c Tc q) ≤c (2c ↑c Tc p)) ↔ ( Nc x ≤c Nc y → (2c ↑c Tc Nc x) ≤c (2c ↑c Tc Nc y))))
56	48, 55	mpbiri 224	. . . . . . . . . . . . . . . . . . 19 ⊢ ((q = Nc x ∧ p = Nc y) → (q ≤c p → (2c ↑c Tc q) ≤c (2c ↑c Tc p)))
57	56	exlimivv 1635	. . . . . . . . . . . . . . . . . 18 ⊢ (∃x∃y(q = Nc x ∧ p = Nc y) → (q ≤c p → (2c ↑c Tc q) ≤c (2c ↑c Tc p)))
58	11, 57	sylbi 187	. . . . . . . . . . . . . . . . 17 ⊢ ((q ∈ NC ∧ p ∈ NC ) → (q ≤c p → (2c ↑c Tc q) ≤c (2c ↑c Tc p)))
59	58	ancoms 439	. . . . . . . . . . . . . . . 16 ⊢ ((p ∈ NC ∧ q ∈ NC ) → (q ≤c p → (2c ↑c Tc q) ≤c (2c ↑c Tc p)))
60	6, 59	sylbird 226	. . . . . . . . . . . . . . 15 ⊢ ((p ∈ NC ∧ q ∈ NC ) → ( Tc q ≤c Tc p → (2c ↑c Tc q) ≤c (2c ↑c Tc p)))
61	60	imp 418	. . . . . . . . . . . . . 14 ⊢ (((p ∈ NC ∧ q ∈ NC ) ∧ Tc q ≤c Tc p) → (2c ↑c Tc q) ≤c (2c ↑c Tc p))
62	61	an32s 779	. . . . . . . . . . . . 13 ⊢ (((p ∈ NC ∧ Tc q ≤c Tc p) ∧ q ∈ NC ) → (2c ↑c Tc q) ≤c (2c ↑c Tc p))
63		breq1 4643	. . . . . . . . . . . . . . . 16 ⊢ (M = Tc q → (M ≤c Tc p ↔ Tc q ≤c Tc p))
64	63	anbi2d 684	. . . . . . . . . . . . . . 15 ⊢ (M = Tc q → ((p ∈ NC ∧ M ≤c Tc p) ↔ (p ∈ NC ∧ Tc q ≤c Tc p)))
65	64	anbi1d 685	. . . . . . . . . . . . . 14 ⊢ (M = Tc q → (((p ∈ NC ∧ M ≤c Tc p) ∧ q ∈ NC ) ↔ ((p ∈ NC ∧ Tc q ≤c Tc p) ∧ q ∈ NC )))
66		oveq2 5532	. . . . . . . . . . . . . . 15 ⊢ (M = Tc q → (2c ↑c M) = (2c ↑c Tc q))
67	66	breq1d 4650	. . . . . . . . . . . . . 14 ⊢ (M = Tc q → ((2c ↑c M) ≤c (2c ↑c Tc p) ↔ (2c ↑c Tc q) ≤c (2c ↑c Tc p)))
68	65, 67	imbi12d 311	. . . . . . . . . . . . 13 ⊢ (M = Tc q → ((((p ∈ NC ∧ M ≤c Tc p) ∧ q ∈ NC ) → (2c ↑c M) ≤c (2c ↑c Tc p)) ↔ (((p ∈ NC ∧ Tc q ≤c Tc p) ∧ q ∈ NC ) → (2c ↑c Tc q) ≤c (2c ↑c Tc p))))
69	62, 68	mpbiri 224	. . . . . . . . . . . 12 ⊢ (M = Tc q → (((p ∈ NC ∧ M ≤c Tc p) ∧ q ∈ NC ) → (2c ↑c M) ≤c (2c ↑c Tc p)))
70	69	com12 27	. . . . . . . . . . 11 ⊢ (((p ∈ NC ∧ M ≤c Tc p) ∧ q ∈ NC ) → (M = Tc q → (2c ↑c M) ≤c (2c ↑c Tc p)))
71	70	rexlimdva 2739	. . . . . . . . . 10 ⊢ ((p ∈ NC ∧ M ≤c Tc p) → (∃q ∈ NC M = Tc q → (2c ↑c M) ≤c (2c ↑c Tc p)))
72	71	3adant1 973	. . . . . . . . 9 ⊢ ((M ∈ NC ∧ p ∈ NC ∧ M ≤c Tc p) → (∃q ∈ NC M = Tc q → (2c ↑c M) ≤c (2c ↑c Tc p)))
73	4, 72	mpd 14	. . . . . . . 8 ⊢ ((M ∈ NC ∧ p ∈ NC ∧ M ≤c Tc p) → (2c ↑c M) ≤c (2c ↑c Tc p))
74	73	3expa 1151	. . . . . . 7 ⊢ (((M ∈ NC ∧ p ∈ NC ) ∧ M ≤c Tc p) → (2c ↑c M) ≤c (2c ↑c Tc p))
75	74	an32s 779	. . . . . 6 ⊢ (((M ∈ NC ∧ M ≤c Tc p) ∧ p ∈ NC ) → (2c ↑c M) ≤c (2c ↑c Tc p))
76		breq2 4644	. . . . . . . . 9 ⊢ (N = Tc p → (M ≤c N ↔ M ≤c Tc p))
77	76	anbi2d 684	. . . . . . . 8 ⊢ (N = Tc p → ((M ∈ NC ∧ M ≤c N) ↔ (M ∈ NC ∧ M ≤c Tc p)))
78	77	anbi1d 685	. . . . . . 7 ⊢ (N = Tc p → (((M ∈ NC ∧ M ≤c N) ∧ p ∈ NC ) ↔ ((M ∈ NC ∧ M ≤c Tc p) ∧ p ∈ NC )))
79		oveq2 5532	. . . . . . . 8 ⊢ (N = Tc p → (2c ↑c N) = (2c ↑c Tc p))
80	79	breq2d 4652	. . . . . . 7 ⊢ (N = Tc p → ((2c ↑c M) ≤c (2c ↑c N) ↔ (2c ↑c M) ≤c (2c ↑c Tc p)))
81	78, 80	imbi12d 311	. . . . . 6 ⊢ (N = Tc p → ((((M ∈ NC ∧ M ≤c N) ∧ p ∈ NC ) → (2c ↑c M) ≤c (2c ↑c N)) ↔ (((M ∈ NC ∧ M ≤c Tc p) ∧ p ∈ NC ) → (2c ↑c M) ≤c (2c ↑c Tc p))))
82	75, 81	mpbiri 224	. . . . 5 ⊢ (N = Tc p → (((M ∈ NC ∧ M ≤c N) ∧ p ∈ NC ) → (2c ↑c M) ≤c (2c ↑c N)))
83	82	com12 27	. . . 4 ⊢ (((M ∈ NC ∧ M ≤c N) ∧ p ∈ NC ) → (N = Tc p → (2c ↑c M) ≤c (2c ↑c N)))
84	83	rexlimdva 2739	. . 3 ⊢ ((M ∈ NC ∧ M ≤c N) → (∃p ∈ NC N = Tc p → (2c ↑c M) ≤c (2c ↑c N)))
85	84	3ad2antl1 1117	. 2 ⊢ (((M ∈ NC ∧ N ∈ NC ∧ (N ↑c 0c) ∈ NC ) ∧ M ≤c N) → (∃p ∈ NC N = Tc p → (2c ↑c M) ≤c (2c ↑c N)))
86	3, 85	mpd 14	1 ⊢ (((M ∈ NC ∧ N ∈ NC ∧ (N ↑c 0c) ∈ NC ) ∧ M ≤c N) → (2c ↑c M) ≤c (2c ↑c N))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   ⊆ wss 3258  ℘cpw 3723  ℘₁cpw1 4136  0_cc0c 4375   class class class wbr 4640  (class class class)co 5526   ≈ cen 6029   NC cncs 6089   ≤_c clec 6090   Nc cnc 6092   T_c ctc 6094  2_cc2c 6095   ↑_c cce 6097

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-compose 5749  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-lec 6100  df-nc 6102  df-tc 6104  df-2c 6105  df-ce 6107

This theorem is used by:  nchoicelem9  6298